Vector differentiation. Chapters 6, 7

Chapter 2

Vectors

Courtesy NASA/JPL-Caltech

Summary

Circa 1900 A.D., J. Williard Gibbs invented a useful combination of magnitude and direction called vec-tors and their higher-dimensional counterparts dyadics, triadics, and polyadics. Vectors are an important geometrical tool e.g., for surveying, motion analysis, lasers, optics, computer graphics, ani-mation, CAD/CAE (computer aided drawing/engineering), and FEA.

Symbol Description Details

0,
u Zero vector and unit vector. Sections 2.3, 2.4

+ − ∗ Vector addition, negation, subtraction, Sections 2.6 - 2.9

and multiplication/division with a scalar.

··· × Vector dot product and cross product. Sections 2.10, 2.11 F_d

dt Vector differentiation. Chapters 6, 7

2.1

Examples of scalars, vectors, and dyadics

• Ascalar is a non-directional quantity(e.g., a real number). Examples include: time density volume mass moment of inertia temperature distance speed angle weight potential energy kinetic energy

• Avector is a quantity that has magnitude and one associated direction. For example, avelocity vector has speed(how fast something is moving)and direction(which way it is going). Aforce vector has magnitude(how hard something is pushed) and direction (which way it is shoved). Examples include:

position vector velocity acceleration linear momentum force impulse angular velocity angular acceleration angular momentum torque

• Adyad is a quantity with magnitude andtwoassociated directions. For example,stress associates with area and force(both regarded as vectors). Adyadic is thesum of dyads. For example, aninertia dyadic (Chapter 14) is the sum of dyads associated with moments and products of inertia.

Words: Vector and column matrices. Although mathematics uses the word1 _{“vector” to describe a column}

matrix, a column matrix doesnot have direction. To associate direction, attach a basis e.g., as shown below.
ax + 2
ay + 3
az = 
ax
ay
az _ 1 2 3   =  1 2 3
a_xyz

where
ax,
ay,
az are orthogonal unit vectors ax

ay

az

Note: Although it can be helpful to represent vectors with orthogonal unit vectors (e.g.,
x,
y,
z or

i,
j,
k), it is not always necessary, desirable, or efficient. Postponing resolution of vectors into components allows maximum use of simplifying vector properties and avoids simplifications such as sin2(θ) + cos2(θ) = 1 (see Homework 2.9).

1

2.2

Definition of a vector

Avector is defined as a quantity having magnitude and direction.a

Vectors are represented graphically with straight or curved arrows(examples below).

right / left up/down out-from / into page inclined at 45o Certain vectors have additional special properties. For example, aposition vector is associated with two points and has units of distance. A bound vector such as force

is associated with a point(or line of action).

Courtesy Bro. Claude Rheaume LaSalette.

a_{A vector’smagnitude is a real non-negative number. A vector’sdirection can be resolved intoorientation andsense.} For example, a highway has an orientation (e.g., east-west) and a vehicle traveling east has a sense. Knowing both the orien-tation of a line and the sense on the line gives direction. Changing a vector’s orienorien-tation or sense changes its direction.

Example of a vector: Consider the traffic report “the vehicle is heading East at 5 m_s”. It isconvenient

to name these two pieces of information(speed and direction) as a “velocity vector” and represent them mathematically as 5∗
East (direction is identified with an arrow and/or bold-face font or with a hat for a unit vector such as
East). The vehicle’s speed is always a real non-negative number, equal to themagnitude

of the velocity vector. The combination of magnitude and direction is avector. For example, the vector
v describing a vehicle traveling with speed 5

to the East is graphically depicted to the right, and is written

v = 5 ∗
East or
v = 5
East

A vehicle traveling with speed 5 to the West is

v = 5 West or
v = -5
East N S E W m sec 5

Note: The negative sign in -5

East is associated with the vector’s direction (the vector’s magnitude is inherently positive). When a vector is written in terms of a scalar x that can be positiveor zero or negative, e.g., as xEast,
the vector’s inherently non-negativemagnitude is abs(x) whereas x is called theEast
measure of the vector.

2.3

Zero vector
0 and its properties

Azero vector
0 is defined as a vector whose magnitude is zero.2

Addition of a vector
v with a zero vector:
v +
0 =
v

Dot product with a zero vector:
v ···
0 = (2) 0

0 isperpendicular to all vectors Cross product with a zero vector:
v ×
0 =

(5)
0 0 is

parallel to all vectors Derivative of the zero vector: Fd
0

dt =
0 F is any reference frame

Vectors a and b are said to be “perpendicular” if a··· b = 0 whereas a and b are “parallel” if a× b = 0.

Note: Some say
a and
b are “parallel” only if
a and
b have the same direction and anti-parallel if
a and
b have opposite directions. 2_{The direction of a zero vector 0 is arbitrary and may be regarded as havinganydirection so that 0 is parallel to all} vectors, 0 isperpendicular to all vectors, all zero vectors are equal, and one may use the definite pronoun “the” instead of the indefinite “a” e.g., “the zero vector”. It is improper to say thezero vector has no direction as a vector isdefined to

2.4

Unit

vectors

A unit vector is defined as a vector whose magnitude is 1, and is designated with a special hat, e.g.,
u. Unit vectors can be “sign posts”, e.g., unit vectors
N,
S, W,
E associated

with local Earth directions North, South, West, East, respectively. The direction of unit vectors are chosen to simplify communication and to produce efficient equations. Other useful “sign posts” are:

• Unit vector directed from one point to another point • Unit vector directed locally vertical

• Unit vector tangent to a curve

• Unit vector parallel to the edge of an object • Unit vector perpendicular to a surface

N

S

E W

A unit vector can be defined so it has the same direction as an arbi-trary non-zero vector
v by dividing
v by |
v| (the magnitude of v). To avoid divide-by-zero problems during numerical computation, approximate the unit vector with a “small” positive real number  in the denominator.

unitVector =
v |
v| ≈ _{|
v| + }
v (1)

2.5

Equal vectors (

= )

Two vectors are “equal” when they have the same magnitude and same direction.a Shown to the right are threeequal vectors. Although each has a different location, the vectors are equal because they have the same magnitude and direction.b

a

Homework 2.6 draws vectors of different magnitude,orientation, andsense.

b_{Some vectors have additional properties. For example, a position vector is associated with two points. Two position vectors} areequal position vectors when, they have the same magnitude, same direction, and are associated with the same points. Two force vectors areequal force vectors when they have the same magnitude, direction, and point of application.

2.6

Vector addition (

+ )

As graphically shown to the right, adding two vectors
a +
b produces a vector.a First, vector
b is translatedb so its tail is at the tip of
a. Next, the vector
a +
b is drawn from the tail of
a to the tip of the translated
b.

Properties of vector addition

Commutative law:
a +
b =
b +
a

Associative law: (
a +
b ) +
c =
a + (
b +
c ) =
a +
b +
c Addition of zero vector:
a +
0 =
a

a

It does not make sense to add vectors with different units, e.g., it is nonsensical to add a velocity vector with units of m

s with an angular velocity vector with units of rad sec. b

Translating b does not change the magnitude or direction of b, and so produces an equal b.

b a

a

a + b b

Example: Vector addition ( + )

Shown to the right is an example of how to add vector
w to vector
v, each which is expressed in

terms of orthogonal unit vectors
nx,
ny,
nz.

v = 7
nx + 5
ny + 4
nz +
w = 2
nx + 3
ny + 2
nz 9
nx + 8
ny + 6
nz

n

n

n

2.7

Vector multiplied or divided by a scalar (

∗ or / )

• Multiplying a vector by a positivenumber (other than 1) changes the vector’s magnitude.

• Multiplying a vector by a negative number

changes the vector’s magnitude andreverses thesense of the vector.

• Dividing a vector
a by a scalar s1 is defined as
a s1

1

s1 ∗
a Properties of multiplication of a vector by a scalar s1 or s2

Commutative law: s1
a =
a s1

Associative law: s1( s2
a ) = ( s1s2)
a = s2( s1
a ) = s1s2
a

Distributive law: (s1+ s2)
a = s1
a + s2
a

Distributive law: s1(
a +
b ) = s1
a + s1
b

Multiplication by zero: 0∗
a =
0

a

Homework 2.9 multiplies a vector b by various scalars.

a

2a

-2a

a

Examples: Vector scalar multiplication and division (∗ or / )

v = 7
nx + 5
ny + 4
nz 5
v = 35
nx + 25
ny + 20
nz
v -2 = -3.5
nx - 2.5
ny - 2
nz

n

n

n

2.8

Vector negation (

− )

A graphical representation of negating a vector
a is shown to the right.

Negating a vector (multiplying the vector by -1) changes the sense of a vector without changing its magnitude or orientation. In other words, multiplying a vector by -1 reverses the sense of the vector (it points in the opposite direction).

a

-a

2.9

Vector subtraction (

− )

As graphically shown to the right, the process of subtracting a vector
b from a vector
a is simply addition and negation,a i.e.,

a −
b

a + -
b

After negating vector
b, it is translated so the tail of -
b is at the tip of
a. Next, the vector
a + -
b is drawn from the tail of
a to the tip of the translated -
b.b

a

In most (or all) mathematical processes, subtraction is defined as negation and addition.

b

Homework 2.11 draws b−a
b +-a.

b a

a + -b a

-_b

Example: Vector subtraction ( − )

Shown to the right is an example of how to subtract vector
w from vector
v, each which is expressed in

v = 7
nx + 5
ny + 4
nz

-
w = 2
nx + 3
ny + 2
nz

n

n

n

2.10

Vector dot product (

··· )

Equation (2) defines thedot product of vectors
a and
b.

• |
a| and |
b| are the magnitudes of
a and
b, respectively. • θ is the smallest angle between
a and
b (0≤ θ ≤ π). Equation (3) is a rearrangement of equation (2) that is useful for calculating the angle θ between two vectors. Note: a and b are “perpendicular” when a··· b = 0.

Note: Dot-products encapsulate thelaw of cosines.

b

a

θ

|a|

|b

|

Use acosacosacosacosacosacosacosacosacosacosacosacosacos to calculate θ.

Equation (2) shows
v···
v = |
v|2. Hence, the dot product can calculate a vector’s magnitude as shown for |
v| in equation (4). Equation (4) also defines vector exponentiation
vn

(vector v raised to scalar power n) as a non-negative scalar. Example: Kinetic energy K = 1₂m
v2 =

(4) 1 2m
v···
v
v2
|
v|2 =
v ···
v |
v| = +√
_v···
v
vn
|
v|n= +_(
v···
v)n2 (4)

2.10.1 Properties of the dot-product (··· )

Dot product with a zero vector
a ···
0 = 0

Dot product of perpendicular vectors
a ···
b = 0 if
a ⊥
b Dot product of parallel vectors
a ···
b = ± |
a| |
b| if
a 
b Dot product with vectors scaled by s1 and s2 s1
a ··· s2
b = s1s2

a···
b

Commutative law
a ···
b =
b ···
a

Distributive law
a ··· (
b +
c ) =
a ···
b +
a ···
c

Distributive law (
a +
b )··· (
c +
d ) =
a ···
c +
a ···
d +
b ···
c +
b ···
d Note: The proofs of the distributive law for dot-products and cross-products is found in [40, pgs. 23-24, 32-34].

2.10.2 Uses for the dot-product (··· )

• Calculating an angle between two vectors [see equation (3) and example in Section 3.3] or determining when two vectors areperpendicular, e.g.,
a···
b = 0.

• Calculating a vector’s magnitude [see equation (4) anddistance examples in Sections 3.2 and 3.3].

• Changing avector equation into a scalar equation (see Homework 2.34).

• Calculating a unit vector in the direction of a vector
v [see equation (1)] unitVector =

(1)

v |
v|

• Projection of a vector
v in the direction of
b is defined:
v···
b

|
b|. See Section 4.2 forprojections,measures,coefficients,components.

2.10.3 Special case: Dot-products with orthogonal unit vectors When
nx,
ny,
nz areorthogonal unitvectors, it can be shown (see Homework 2.4)

(ax
nx + ay
ny + az
nz)··· (bx
nx + by
ny + bz
nz) = axbx + ayby + azbz

n

n

n

Optional: Special case of dot product as matrix multiplication

When one defines
a
ax
nx + ay
ny + az
nz and
b
bx
nx + by
ny + bz
nz in terms of the orthog-onal unit vectors
nx,
ny,
nz, the dot-product
a ···
b is related to the multiplication of the
nx,
ny,
nz

row matrix representation of
a with the
nx,
ny,
nz column matrix representation of
b as
a ···
b = ax ay az 
nxyz ∗    bx by bz   
nxyz = axbx + ayby + azbz 

n

n

n

The following shows how to use dot-products with the vectors
v and
w, each which is

expressed in terms of the orthogonal unit vectors
nx,
ny,
nz shown to the right.
v = 7
nx + 5
ny + 4
nz
w = 2
nx + 3
ny + 2
nz

n

n

n

v ···
v = 72 _{+ 5}2 _{+ 4}2 _{= 90 |
v| =} √_{90 ≈ 9.4868}

w ···
w = 22 + 32 + 22 = 17 |
w| = √17 ≈ 4.1231 Unit vector in the direction of
v:
v

|
v| =

7
nx + 5_√
ny + 4
nz

90 ≈ 0.738
nx + 0.527
ny + 0.422
nz

Unit vector in the direction of
w:
w |
w| = 2
nx + 3_√
ny + 2
nz 17 ≈ 0.485
nx + 0.728
ny + 0.485
nz
v···
w = 7 ∗ 2 + 5 ∗ 3 + 4 ∗ 2 = 37 ∠∠∠∠∠∠∠∠∠∠∠∠∠
v,
w = acos√ 37 90 √17  ≈ 0.33 rad ≈ 18.93◦

2.10.5 Dot-products to change vector equations to scalar equations (see Hw 1.34)

One way to form up to three linearly independent scalar equations from the vector equation

v =
0 is by dot-multiplying
v =
0 with three orthogonal unit vectors
a1,
a2,
a3, i.e.,

if
v =
0 ⇒
v ···
a1 = 0
v ···
a2 = 0
v ···
a3 = 0

a

a

a

2.11

Vector cross product (

× )

Thecross product of a vector
a with a vector
b is defined in equation (5).

• |
a| and |
b| are the magnitudes of
a and
b, respectively • θ is the smallest angle between
a and
b (0≤ θ ≤ π).

•
u is the unit vector perpendicularto both
a and
b. The direction of
u is determined by theright-hand rule.a

Note:|a| |b| sin(θ) [the coefficient ofu in equation (5)] is inherently
non-negative since sin(θ)≥ 0 because 0 ≤ θ ≤ π. Hence, |a ×b| = |a| |b| sin(θ).

a

The right-hand rule is a convention, much like driving on the right-hand side of the road in North America. Until 1965, the Soviet Union used the left-hand rule.

b

a

θ

|a|

|b

|

u

2.11.1 Properties of the cross-product (× ) Cross product with a zero vector
a ×
0 =
0

Cross product of a vector with itself
a ×
a =
0

Cross product of parallel vectors
a ×
b =
0 if
a 
b Cross product with vectors scaled by s1 and s2 s1
a × s2
b = s1s2

a ×
b

Cross products arenot commutative
a ×
b = -
b ×
a (6) Distributive law
a × 
b +
c =
a ×
b +
a ×
c

Distributive law 
a +
b×
c +
d =
a×
c +
a ×
d +
b ×
c +
b ×
d Cross products arenot associative
a × 
b ×
c = 
a×
b×
c

Vector triple cross product
a×
b ×
c =
b
a···
c −
c
a···
b (7) A mnemonic for a× (b ×c) = b (a ···c) − c (a ··· b) is “backcab” - as in were you born in the back of a cab?

Many proofs of this formula resolve a, b, and c into orthogonal unit vectors (e.g.,n
x,n
y,
nz) and equate components.

2.11.2 Uses for the cross-product (× )

Several uses for the cross-product in geometry, statics, and motion analysis, include calculating:

• Perpendicular vectors, e.g.,
a×
b is perpendicular to both
a and
b

• Moment of a force or linear momentum, e.g.,
r×
F and
r× m
v

• Velocity/acceleration formulas, e.g.,
v =
ωωωωωωωωωωωωω×
r and
a =
ααααααααααααα×
r +
ωωωωωωωωωωωωω× (
ωωωωωωωωωωωωω×
r) • Area of a triangle whose sides have length |
a| and |
b|

b

a

θ

|b

|

|a|

The area of a triangle ∆ is half the area of a parallelogram.a b A geometrical interpretation of |
a ×
b| is thearea of a parallelogram

having sides of length |
a| and |
b|, hence

∆(
a,
b) = 1₂|
a ×
b| (8)

a_{Homework 2.14 shows the utility of equation (8) forsurveying.} b_{Section 3.3 shows the utility of a cross-product for area calculations.} • Distance d between a line L and a point Q.

P

u

d

Q

r

θ

L

d = 
rQ/P ×
u =

(5)


rQ/P sin(θ) (9) Note: See example in Hw 1.28. Other distance calculations are in Sections 3.2 and 3.3.

• The vector
b⊥ (shown right)is perpendicular to
b and is in the plane containing both
a and
b. It is calculated with thevector triple cross product:

b_⊥ = 
a×
b×
b In general,  b ⊥   =  b  

and b⊥is not perpendicular to a.

b

a

bT

2.11.3 Special case: Cross-products with right-handed, orthogonal, unit vectors When
nx,
ny,
nz areorthogonal unit vectors, it can be shown (see Homework 2.13)that

the cross product of
a = ax
nx + ay
ny + az
nz with
b = bx
nx + by
ny + bz
nz

happens to be equal to the determinant of the following matrix.

n

n

n

2.11.4 Examples: Vector cross-products (× )

The following shows how to use cross-products with the vectors
v and
w, each which

is expressed in terms of the orthogonal unit vectors
nx,
ny,
nz shown to the right.
v = 7
nx + 5
ny + 4
nz
w = 2
nx + 3
ny + 2
nz

n

n

n

Area from vectors
v and
w: ∆(
v,
w) = 1

2|
v ×
w| = 1 2  -22 _{+ -6}2 _{+ 11}2 ₌ √ 161 2 ≈ 6.344
v× (
v ×
w) = det   
nx
ny
nz 7 5 4 -2 -6 11    = 79
nx − 85
ny − 32
nz

2.12

Optional: Scalar triple product (

··· × or × ··· )

The scalar triple product of vectors
a,
b,
c is the scalar defined in the various ways shown in equation (10).3

ScalarTripleProduct

a ···
b ×
c =
a ×
b ···
c =
b ···
c ×
a =
b ×
c ···
a (10) 2.12.1 Scalar triple product and the volume of a tetrahedron

A geometrical interpretation of
a ···
b ×
c is thevolume of a parallelepiped having sides of length |
a|, |
b|, and |
c|. The formula for thevolume of a tetrahedron whose sides are described by the vectors
a,
b,
c is

Tetrahedron Volume = 1

6
a ···
b ×
c

This formula is used for volume calculations (e.g., highway surveying cut

and fill), 3DCAD, solid geometry, and mass property calculations. a b c

2.12.2 ( × ··· ) to change vector equations to scalar equations (see Hw 1.34)

Section 2.10.5 showed one method to form scalar equations from the vector equation
v =
0. A 2nd method expresses
v in terms of three non-coplanar (but not necessarily orthogonal or unit) vectors
a1,
a2,
a3, and writes the equally valid (but generally different) set of linearly

independent scalar equations shown below.

Method 2: if
v = v1
a1 + v2
a2 + v3
a3 =
0 ⇒ v1= 0 v2 = 0 v3 = 0 Note: The proof that vi= 0 (i = 1, 2, 3) follows directly by substituting v = 0 into equation (4.2).

a3

a2

a1

Vectors are used withsurveying data for volume cut-and-fill dirt calculations for highway construction 3

Although parentheses make equation (10) clearer, i.e., ScalarTripleProduct
a ··· (b ×c), the parentheses are unnecessary because the cross product b×c mustbe performed before the dot product for a sensible result to be produced.